4.1. Weak magnetic fields

Each Dirac fermion has a spin and in the presence of an external magnetic field the spins of these fermions are polarized and a net magnetic moment arises (the Pauli paramagnetism). Besides, when a gas consisting of massless fermions is placed in a magnetic field, the motion of the fermions changes: they begin to move along the helical trajectories. Hence, these magnetized Dirac fermions generate an additional magnetic field, whose direction is opposite to that of the external field. This is the Landau diamagnetism effect due to the variation of the orbital motion of fermions caused by the external field. This effect is of a quantum nature and is absent in the classical approximation. In fact, the diamagnetism occurs and results from the quantization of massless fermions levels in the magnetic field (see § 3). We conclude that the magnetization of a Dirac plasma of massless fermions results from the Pauli paramagnetism and the Landau diamagnetism.

Let us calculate the thermodynamic potential (Landau & Lifshitz Reference Landau and Lifshitz1980)

(4.1)\begin{equation} \varOmega ={-}\beta^{{-}1}\sum_n \ln \left[1 + \exp \left(\eta - \beta \epsilon_n\right)\right] , \end{equation}

where $\beta ^{-1} = T$ is the system temperature in energy units (throughout the article we put the Boltzmann constant $k_{\textrm {B}} = 1$), and the sum is taken over all possible states, in the case of weak magnetic fields, i.e. when the inequality $\beta \hbar \omega _{\textrm {H}} \ll 1$ holds. The dimensionless chemical potential $\eta$ in (4.1), in 3-D plasmas, is determined by the condition (D2). Introducing the density of states in accordance with (3.10), we obtain

(4.2)\begin{equation} \varOmega_{{\pm}} ={-}\frac{V}{2{\rm \pi}^2 \hbar}\frac{eH}{\beta c}\int_0^{\infty}\,{d}\epsilon \sum_{n}\frac{{d}k_{z}}{{d}\epsilon}\ln \left\{1 + \exp \left[\eta -\beta \epsilon_{n,\pm}(k_z) \right]\right\} . \end{equation}

At small $H$ we may replace the summation by integration using the Euler–Maclaurin sum formula (2.1) in the Supplemental material. In the case of a degenerate Dirac plasma ($\beta \epsilon _{\textrm {F}} \gg 1$), for the right helicity ($+$), we put in equation (2.1) in the Supplemental material $a = 0$, $b = \epsilon _{\textrm {F}}^2/(\hbar ^2 \omega _{\textrm {H}}^2) - 1$, and for the left helicity ($-$) we apply $a = 1$, $b = \epsilon _{\textrm {F}}^2/(\hbar ^2 \omega _{\textrm {H}}^2)$, where $\epsilon _{\textrm {F}} = \hbar v k_{\textrm {F}}$ is the Fermi energy with $k_{\textrm {F}} = \sqrt [3]{3{\rm \pi} ^2 n_3}$ ($n_3$ is the 3-D Dirac fermion density). As a result, we obtain

(4.3a–c)\begin{equation} \varOmega_{{\pm}} ={-}\frac{V}{8{\rm \pi}^2}\frac{\epsilon_{{\rm F}}^2 \omega_{{\rm H}}^2}{\hbar v^3}S_{{\pm}} , \quad S_+{=}\sum_{n=0}^{[n_{0,+}]}f_+(n) ,\quad S_{-} = \sum_{n=1}^{[n_{0,-}]} \left[1 + f_{-}(n)\right] , \end{equation}

where $[x]$ is the integer part of number $x$,

(4.4)\begin{equation} \left. \begin{aligned} f_+(n) & = \sqrt{\frac{n_{0,+} - n}{n_{0,+} + 1}} - \frac{n+1}{n_{0,+} + 1} \ln\left(\sqrt{\frac{n_{0,+} + 1}{n+1}} + \sqrt{\frac{n_{0,+} - n}{n+1}}\right) , \\ f_{-}(n) & = \sqrt{1 - \frac{n}{n_{0,-}}} - \frac{n}{n_{0,-}} \ln\left(\sqrt{\frac{n_{0,-}}{n}} + \sqrt{\frac{n_{0,-}}{n} - 1}\right) \end{aligned} \right\} \end{equation}

and

(4.5a,b)\begin{equation} n_{0,+} = \epsilon_{{\rm F}}^2/(\hbar^2 \omega_{{\rm H}}^2) - 1 , \quad n_{0,-} = \epsilon_{{\rm F}}^2/(\hbar^2 \omega_{{\rm H}}^2) . \end{equation}

The magnetic moment per unit volume is defined as

(4.6)\begin{equation} \boldsymbol{\mathfrak{M}}={-}\frac{1}{V}\left(\frac{\partial \varOmega}{\partial \boldsymbol{H}}\right)_{\beta ,V,\eta}. \end{equation}

From (4.3a–c) and (4.4), using the Euler–Maclaurin formula (2.1) in the Supplemental material, after some algebra, in the limit $H \rightarrow \boldsymbol {0}$ we arrive at the expression for the 3-D magnetic moments

(4.7)\begin{equation} \boldsymbol{\mathfrak{M}}_{{\rm F},\pm}\simeq{\mp} \boldsymbol{\mathfrak{m}}_{{\rm F}} + \frac{5\alpha}{48{\rm \pi}^2} \frac{v}{c}\left[1-\frac{4}{5}\ln \left(\frac{\hbar \omega_{{\rm H}}}{2\epsilon_{F}}\right)\right] \boldsymbol{H}, \end{equation}

where $\alpha = e^2/(\hbar c)\simeq 1/137$ is the fine structure constant and

(4.8)\begin{equation} |\boldsymbol{\mathfrak{m}}_{{\rm F}}|=\frac{e}{8{\rm \pi}^2}\frac{\epsilon_{{\rm F}}^2}{\hbar^2 vc} \end{equation}

is the proper magnetic moment of the degenerate Dirac plasma. Then the total magnetic moment of the degenerate massless Dirac plasma is

(4.9)\begin{equation} \boldsymbol{\mathfrak{M}}_{{\rm F}, {\rm tot}} = \boldsymbol{\mathfrak{M}}_{{\rm F},+} + \boldsymbol{\mathfrak{M}}_{{\rm F},-}\simeq \frac{5\alpha}{24{\rm \pi}^2}\frac{v}{c}\left[1-\frac{4}{5} \ln \left(\frac{\hbar \omega_{{\rm H}}}{2\epsilon_{{\rm F}}}\right)\right]\boldsymbol{H} . \end{equation}

In addition, for the (linear) magnetic susceptibility of degenerate Dirac plasmas we obtain

(4.10)\begin{equation} \chi_{{\rm F},\pm} \simeq \frac{\alpha}{16{\rm \pi}^2}\frac{v}{c} \left[1-\frac{4}{3}\ln \left(\frac{\hbar \omega_{{\rm H}}}{2\epsilon_{\rm F}}\right)\right] \end{equation}

and

(4.11)\begin{equation} \chi_{{\rm F}, {\rm tot}}\simeq \frac{\alpha}{8{\rm \pi}^2}\frac{v}{c}\left[1 - \frac{4}{3} \ln \left(\frac{\hbar \omega_{\rm H}}{2\epsilon_{\rm F}}\right)\right] . \end{equation}

In the opposite case of high temperatures ($\beta \epsilon _\textrm {F}\ll 1$) the Dirac fermions form a Boltzmann gas, and the magnetic moments are

(4.12)\begin{equation} \boldsymbol{\mathfrak{M}}_{{\rm B},\pm} \simeq{\mp} \boldsymbol{\mathfrak{m}}_{\rm B} + \frac{5ne}{64} \frac{\omega_{\rm H}}{c}l^{2}\left(1 - \frac{4}{15}\beta \hbar \omega_{\rm H}\right) \frac{\boldsymbol{H}}{H}, \end{equation}

where

(4.13)\begin{equation} |\boldsymbol{\mathfrak{m}}_{\rm B}|=\frac{lne}{8}\frac{v}{c} \end{equation}

is the proper magnetic moment of a Dirac plasma in the case of the Boltzmann statistics. The total magnetic moment of a Boltzmann massless Dirac plasma reads

(4.14)\begin{equation} \boldsymbol{\mathfrak{M}}_{{\rm B}, {\rm tot}} \simeq \frac{5ne}{32}\frac{\omega_{\rm H}}{c}l^2 \left(1-\frac{4}{15}\beta \hbar \omega_{\rm H}\right)\frac{\boldsymbol{H}}{H}. \end{equation}

In turn, for the magnetic susceptibilities we have

(4.15)\begin{equation} \chi_{{\rm B},\pm }\simeq \frac{15}{64{\rm \pi} \beta }\frac{l^{2}\omega_{p}^{2}}{\hbar \omega_{\rm H}c^{2}} \left(1-\frac{8}{15}\beta \hbar \omega_{\rm H}\right) \end{equation}

and

(4.16)\begin{equation} \chi_{{\rm B}, {\rm tot}}\simeq \frac{15}{32{\rm \pi} \beta }\frac{l^{2}\omega_{p}^{2}} {\hbar \omega_{\rm H}c^{2}}\left(1 - \frac{8}{15}\beta \hbar \omega_{\rm H}\right) , \end{equation}

where $l = \beta \hbar v$, $\omega _p = \sqrt {4{\rm \pi} n_3 e^2/\mu }$ is the plasma frequency and $\mu = 3/(\beta v^2 )$ is the effective Dirac ‘mass’. As it stems from (4.7) and (4.12), the massless plasma, consisting of fermions with only positive or only negative helicity, possesses the proper magnetic moment in the absence of an external magnetic field. In other words, it possesses its own magnetization.

4.2. Strong magnetic fields

In the overdense plasma (e.g. at the critical electron number density $n_e \sim 10^{21}$ $\textrm {cm}^{-3}$), under the influence a laser pulse with the wavelength $\sim 1.054\,\mathrm {\mu }\textrm {m}$, of duration $\tau _0 \sim 1$ ps and with a high-intensity $\sim 10^{20}\,\textrm {W}\,\textrm {cm}^{-2}$, were generated ultrastrong magnetic fields in the range 50–700 MG. In these experiments (Tatarakis et al. Reference Tatarakis, Watts, Beg, Clark, Dangor, Gopal, Haines, Norreys, Wagner, Wei, Zepf and Krushelnick2002a,Reference Tatarakis, Gopal, Watts, Beg, Wei, Dangor, Krushelnick, Wagner, Norreys, Clark, Zepf and Evans2002b; Wagner et al. Reference Wagner, Tatarakis, Gopal, Beg, Clark, Dangor, Evans, Haines, Mangles and Norreys2004; Sarri et al. Reference Sarri, Macchi, Cecchetti, Kar, Liseykina, Yang, Dieckmann, Fuchs, Galimberti and Gizzi2012) atoms in the laser plasma are ionized in the tunnelling limit ($\gamma \ll 1$) for a time $\tau \sim 1$ fs ($\tau \ll \tau _0$), in the barrier suppression regime.

Let us now consider strong magnetic fields under the condition $1\lesssim \beta \hbar \omega _\textrm {H} \ll |\eta |$. In this case the magnetization of the Dirac plasma contains a contribution which oscillates with a large amplitude as a function of $\boldsymbol {H}$, DHVA effect (Landau & Lifshitz Reference Landau and Lifshitz1980). To separate the oscillatory parts of the thermodynamic potential (4.1) it is convenient to transform the sum in (4.2) by means of the Poisson formula (2.2) in the Supplemental material. In the 3-D case this leads to

(4.17)\begin{gather} \varOmega_{{\pm}} = \varOmega_{0,\pm}(\mu) + \frac{V}{{\rm \pi}^{2}\hbar}\frac{eH}{\beta c}{\rm Re} \sum_{n=1}^{\infty }I_{n}, \end{gather}

(4.18)\begin{gather} I_n ={-}\int_{-\infty}^{\infty}\int_0^{\infty}\ln \left[1+\exp \left(\eta -\beta \epsilon_{x,\pm}(k_z)\right)\right]\exp \left(2{\rm \pi} {\rm i}nx\right)\,{\rm d}x\,{\rm d}k_z, \end{gather}

where $\varOmega _{0,\pm }(\mu )$ is the thermodynamic potential in the absence of the field. In the integrals $I_n$ the variable $x$ is taken to be $x = (\epsilon ^2 - \hbar ^2\omega _\textrm {H}^2 - v^2\hbar ^2 k_z^2)/ (\hbar ^2\omega _\textrm {H}^2)$ for the right-handed helicity and $x = (\epsilon ^2 - v^2\hbar ^2 k_z^2)/ (\hbar ^2\omega _\textrm {H}^2)$ for the left-handed helicity. Then, for the required oscillatory part of the integrals we have

(4.19)\begin{equation} I_{\text{osc}, n} ={-}\int_{-\infty }^{\infty}\int_{0}^{\infty}\ln \left[1 + \exp \left(\eta - \beta \epsilon \right)\right]\exp \left(\frac{2{\rm \pi} in\epsilon^2} {\hbar^2 \omega_{\rm H}^2}\right)\exp \left(-\frac{2{\rm \pi} in v^2 k_z^2}{\omega_{\rm H}^2}\right) \,{\rm d}\epsilon \,{\rm d}k_z . \end{equation}

The main contribution to the integral over $k_z$ proceeds from the values of $k_z \sim \omega _\textrm {H}/v$ (Landau & Lifshitz Reference Landau and Lifshitz1980). On the other hand, the oscillatory part of the integral is produced by the values of $\epsilon$ near $\eta /\beta$ (see below); the lower limit of integration over $\epsilon$ is therefore taken to be equal to zero. The integration over $k_z$ is separable and is carried out by means of the Poisson integral (2.4) in the Supplemental material. The remaining integral is taken by parts and as a result we obtain

(4.20)\begin{equation} \varOmega_{\text{osc},\pm} = \frac{\sqrt{2}V}{16{\rm \pi}^{3}\beta}\frac{\omega_{\rm H}^{3}}{v^{3}} \sum_{n=1}^{\infty}\frac{1}{n^{3/2}}\int_{0}^{\infty}\frac{\cos \left(\dfrac{2{\rm \pi} y^{2}n} {\beta^2\hbar^2\omega_{\rm H}^2} + \dfrac{\rm \pi}{4}\right)}{\exp(\kern-0.01em y - \eta ) + 1}\,{{\rm d} y}, \end{equation}

where the dimensionless chemical potential $\eta$ is determined by the condition (D2). If the inequality holds $\eta \gg 1$, we make in this integral the change of variable $y - \eta = \xi$, and using the value of the complex integral $J_1$ in equation (2.5) in the Supplemental material, from (4.20), we finally get for the oscillatory part of $\varOmega _{\pm }$,

(4.21)\begin{equation} \varOmega_{\text{osc},\pm}\simeq \dfrac{\sqrt{2}V}{16{\rm \pi}^2\beta} \dfrac{\omega_{\rm H}^3}{v^3}\sum_{n=1}^{\infty}\dfrac{\sin \left(\dfrac{2{\rm \pi} \eta^2 n} {\beta^2\hbar^2\omega_{\rm H}^2} + \dfrac{\rm \pi}{4}\right)}{n^{3/2} \sinh \left[4{\rm \pi}^2n\eta /(\beta^2\hbar^2\omega_{\rm H}^2)\right]}. \end{equation}

In the case if $\eta \lesssim 1$, the integral in (4.20) can be calculated only numerically.

The main contribution to the magnetic moment comes from the differentiation of the most rapidly varying factors in (4.21), i.e. the sines in the numerators. This leads to

(4.22)\begin{equation} \mathfrak{M}_{{\rm osc}, \pm} \simeq \dfrac{e}{2\sqrt{2}{\rm \pi}}\dfrac{\eta^{2}V}{\beta^{3}\hbar^{3} \omega_{\rm H} vc}\sum_{n=1}^{\infty}\dfrac{\cos \left(\dfrac{2{\rm \pi} \eta^{2}n}{\beta^{2}\hbar^{2} \omega_{\rm H}^{2}} + \dfrac{\rm \pi}{4}\right)}{\sqrt{n}\sinh \left[4{\rm \pi}^{2}n\eta /(\beta^{2}\hbar^{2} \omega_{\rm H}^{2})\right]}. \end{equation}

For 2-D Dirac plasmas, in view of equation (2.2) in the Supplemental material,

(4.23)\begin{equation} \varOmega_{{\pm}} = \varOmega_{0,\pm}(\mu) + \dfrac{2S}{\rm \pi}\dfrac{eH}{\beta c}{\rm Re} \sum_{n=1}^{\infty}I_n \end{equation}

and the oscillatory parts of the thermodynamic potential are

(4.24)\begin{equation} I_{\text{osc},n} ={-}\int_0^{\infty}\ln \left[1 + \exp \left(\eta - \beta \epsilon \right) \right]\exp \left(\frac{2{\rm \pi} in\epsilon^2}{\hbar^2 \omega_{\rm H}^2}\right)\,{\rm d}\epsilon . \end{equation}

The integral in (4.24) is taken by parts and then

(4.25)\begin{equation} \varOmega_{\text{osc},\pm} = \dfrac{\hbar S}{4{\rm \pi}^2\beta}\dfrac{\omega_{\rm H}^2}{v^2} \sum_{n=1}^{\infty}\dfrac{1}{n}\int_0^{\infty}\dfrac{\cos \left(\dfrac{2{\rm \pi} y^2 n} {\beta^2\hbar^2\omega_{\rm H}^2}\right)}{\exp(y - \eta) + 1}\,{{\rm d} y}, \end{equation}

where the dimensionless chemical potential $\eta$ is determined by the condition (D12). If the inequality holds $\eta \gg 1$, we make in this integral the change of variable $y-\eta = \xi$, and in view of equation (2.5) in the Supplemental material, we obtain

(4.26)\begin{equation} \varOmega_{\text{osc},\pm}\simeq \dfrac{\hbar S}{4{\rm \pi}^2\beta}\dfrac{\omega_{\rm H}^2}{v^2} \sum_{n=1}^{\infty}\dfrac{\sin \left(\dfrac{2{\rm \pi} \eta^2 n}{\beta^2\hbar^2\omega_{\rm H}^2}\right)} {n\sinh \left[4{\rm \pi}^2 n\eta /(\beta^2\hbar^2\omega_{\rm H}^2)\right]} . \end{equation}

Finally, this leads to the following result for the oscillatory part of the magnetic moment in the 2-D case:

(4.27)\begin{equation} \mathfrak{M}_{{\rm osc},\pm} \simeq \dfrac{e\eta^{2}S}{\beta^3\hbar^2\omega_{\rm H}^2 c} \sum_{n=1}^{\infty}\dfrac{\cos \left(\dfrac{2{\rm \pi} \eta^2 n}{\beta^2\hbar^2\omega_{\rm H}^2}\right)} {\sinh \left[4{\rm \pi}^2 n\eta /(\beta^2\hbar^2\omega_{\rm H}^2)\right]}. \end{equation}

The functions in (4.22) and (4.27) oscillate with high frequency, which is a manifestation of the DHVA effect in 3-D and 2-D massless Dirac plasmas. Its ‘period’ in the variable $1/H$ is

(4.28)\begin{equation} \Delta \left(1/H\right) = 2e\dfrac{\beta^2\hbar v^2}{c\eta^2} \end{equation}

and it increases with the Dirac fermion speed as $v^{2}$. The dimensionless chemical potential $\eta$ in (4.28) is determined by the conditions (D2) and (D12). In the case of non-degenerate hot Dirac plasmas we have constructed analytical approximations (D10) and (D18) for the dimensionless chemical potential $\eta$. As it can be seen from the above calculations the DHVA effect is a direct consequence of the Landau diamagnetism.

Some results of numerical calculations for the value of the normalized magnetic moments $\mathfrak {M}^{\ast }$ (3-D) and $\mathfrak {M}_\textrm {2D}$ (2-D) of Dirac plasmas versus the magnetic field $H$ for various densities and temperatures are presented in figures 1 and 2. The magnitude of the magnetic field $H$ is measured in megagauss (MG), and $n_3$ and $n_2$ are the 3-D and 2-D Dirac fermion density, respectively. The dimensionless chemical potential $\eta$ for these figures is determined by solving (D2) and (D12) numerically. These figures clearly demonstrate the DHVA effect in magnetized massless 3-D and 2-D Dirac plasmas.

Figure 1. The function $\mathfrak {M}^{*}$ versus $H$ for $n_3 = 10^{22}$ cm$^{-3}$, $T = 10$ K and $v \approx c/300$.

Figure 2. The function $\mathfrak {M}_\textrm {2D}$ versus $H$ for $n_2 = 10^{18}$ cm$^{-2}$, $T = 100$ K and $v \approx c/300$.

5.1. The Kubo linear reaction theory

Let a system consisting of interacting massless fermions be in a state of statistical equilibrium, which is described by the Gibbs statistical operator (Akhiezer & Peletminskii)

(5.1)\begin{equation} \hat{w} = \exp \{\varOmega -\beta \hat{\mathcal{H}}_0 - \eta \hat{N})\} , \end{equation}

where $\hat {\mathcal {H}}_0$ is the Hamiltonian of interacting Dirac fermions and $\hat {N}$ is the particle number operator. The Gibbs operator (5.1) acts in the Hilbert space of vectors describing the quantum states of massless fermions in a magnetic field. At some time $t_0$ the external field is switched on, so that the Hamiltonian operator of the system becomes

(5.2)\begin{equation} \hat{\mathcal{H}}(t) = \hat{\mathcal{H}}_0 + \hat{V}^{\rm ext}(t), \end{equation}

where

(5.3)\begin{equation} \hat{V}^{\rm ext}(t) ={-}\dfrac{1}{c}\int \,{d}\boldsymbol{r}\boldsymbol{A}^{\rm ext}(\boldsymbol{r},t)\boldsymbol{\cdot}\hat{\boldsymbol{j}}(\boldsymbol{r}) \end{equation}

is the interaction Hamiltonian in the Weyl gauge with an external electromagnetic field characterized by the vector potential $\boldsymbol {A}^\textrm {ext}(\boldsymbol {r},t)$ and the current-density operator $\hat {\boldsymbol {j}}(\boldsymbol {r})$ is determined by (5.25a,b). The statistical operator $\hat {\rho }(t)$ of the system of Dirac fermions satisfies the Liouville equation

(5.4)\begin{equation} {\rm i}\hbar \dfrac{\partial \hat{\rho}(t)}{\partial t}=\left[\hat{\mathcal{H}}(t), \hat{\rho}(t)\right] , \end{equation}

where $[\hat {A},\hat {B}]$ is the commutator of the operators $\hat {A}$ and $\hat {B}$. In order to find the statistical operator of the system $\hat {\rho }(t)$ for $t > t_0$, we introduce the operator

(5.5)\begin{equation} \tilde{\hat{\rho}}(t) = \exp({{\rm i}\hat{\mathcal{H}}_0 t/\hbar})\hat{\rho}(t) \exp({-{\rm i}\hat{\mathcal{H}}_0 t/\hbar}) \end{equation}

in the interaction picture. Then according to (5.4) we obtain for $\tilde {\hat {\rho }}(t)$ the following equation:

(5.6a,b)\begin{equation} {\rm i}\hbar \dfrac{\partial \tilde{\hat{\rho}}(t)}{\partial t} = \left[\tilde{\hat{V}}^{\rm ext}(t),\tilde{\hat{\rho}}(t)\right] ,\quad \tilde{\hat{V}}^{\rm ext}(t) = \exp({{\rm i}\hat{\mathcal{H}}_0 t/\hbar})\hat{V}^{\rm ext}(t) \exp({-{\rm i}\hat{\mathcal{H}}_0 t/\hbar}) . \end{equation}

We assume that for $t=-\infty$ there was no external field and the system was in a statistical equilibrium state, i.e. $\hat {\rho }(-\infty )=\hat {w}$. Since $[\hat {\mathcal {H}}_0,\hat {w}] = 0$, then $\tilde {\rho }(-\infty )=\hat {w}$. The last equality is the initial condition for (5.6a,b). Then from (5.6a,b) we obtain the integral equation for the operator $\tilde {\rho }(t)$,

(5.7)\begin{equation} \tilde{\hat{\rho}}(t) = \hat{w}-\dfrac{{\rm i}}{\hbar}\int_{-\infty}^{t}\,{\rm d}t^{^{\prime}} \left[\tilde{\hat{V}}^{\rm ext}(t^{^{\prime}}), \tilde{\hat{\rho}}(t^{^{\prime}})\right]. \end{equation}

In the linear approximation for the interaction with the external field, we have

(5.8)\begin{equation} \tilde{\hat{\rho}}(t)\simeq \hat{w}-\dfrac{{\rm i}}{\hbar}\int_{-\infty}^{t}\,{\rm d}t^{^{\prime}} \left[\tilde{\hat{V}}^{\rm ext}(t^{^{\prime}}), \hat{w}\right] . \end{equation}

Let us define the average value of the induced current-density in the system of massless fermions as

(5.9)\begin{equation} \bar{\boldsymbol{j}}(\boldsymbol{r},t) = \mathrm{Tr}\left\{\hat{\rho}(t)\hat{\boldsymbol{j}}(\boldsymbol{r})\right\} . \end{equation}

Taking into account that in a statistical equilibrium the average current is equal to zero, from (5.8) we obtain

(5.10)\begin{equation} \bar{\boldsymbol{j}}(\boldsymbol{r},t) = \dfrac{{\rm i}}{\hbar}\int_{-\infty}^{t}\,{\rm d}t^{^{\prime}}{\rm Tr} \left\{\hat{w}\left[\tilde{\hat{V}}^{\rm ext}(t^{^{\prime}}), \tilde{\hat{\boldsymbol{j}}}(\boldsymbol{r},t)\right]\right\} , \end{equation}

where

(5.11)\begin{equation} \tilde{\hat{\boldsymbol{j}}}(\boldsymbol{r},t) = \exp({{\rm i}\hat{\mathcal{H}}_0 t/\hbar})\hat{\boldsymbol{j}}(\boldsymbol{r}) \exp({-{\rm i}\hat{\mathcal{H}}_0 t/\hbar}) . \end{equation}

In view of (5.3), the $\mu$ – component of the average current is

(5.12)\begin{equation} \overline{j_{\mu}}(\boldsymbol{r},t) ={-}\dfrac{1}{c}\int_{-\infty}^{\infty}\,{\rm d}t^{^{\prime}} \int \,{\rm d}\boldsymbol{r}^{^{\prime}}G_{\mu \nu }(\boldsymbol{r}-\boldsymbol{r}^{^{\prime}}, t-t^{^{\prime}}) A_{\nu}^{\rm ext}(\boldsymbol{r}^{^{\prime}},t^{^{\prime}}) , \end{equation}

where the Green function current–current $G_{\mu \nu }$ is defined as

(5.13)\begin{equation} G_{\mu \nu}(\boldsymbol{r},t) ={-}\dfrac{{\rm i}}{\hbar}\theta (t){\rm Tr} \left\{w\left[j_{\mu}(\boldsymbol{r},t), j_{\nu}(0)\right]\right\} , \end{equation}

where $\theta (t)$ is the Heaviside step function. Passing in (5.12) to the Fourier components according to equation (2.6) in the Supplemental material, we have

(5.14)\begin{equation} \overline{j_{\mu}}(\boldsymbol{k},\omega) ={-}\frac{1}{c}G_{\mu \nu}(\boldsymbol{k},\omega) A_{\nu}^{\rm ext}(\boldsymbol{k},\omega) . \end{equation}

Taking into account that at $\varphi (\boldsymbol {r},t) = 0$

(5.15)\begin{equation} \boldsymbol{E}^{\rm ext}(\boldsymbol{r},t) ={-}\frac{1}{c}\frac{\partial \boldsymbol{A}^{\rm ext}(\boldsymbol{r},t)}{\partial t} , \end{equation}

or for the Fourier components,

(5.16)\begin{equation} \boldsymbol{E}^{\rm ext}(\boldsymbol{k},\omega) = {\rm i}\frac{\omega}{c}\boldsymbol{A}^{\rm ext}(\boldsymbol{k}, \omega), \end{equation}

(5.14) can be rewritten as

(5.17)\begin{equation} \overline{j_{\mu}}(\boldsymbol{k},\omega) = \frac{{\rm i}}{\omega}G_{\mu \nu}(\boldsymbol{k},\omega) E_{\nu}^{\rm ext}(\boldsymbol{k},\omega). \end{equation}

Then, according to the definition

(5.18)\begin{equation} \overline{j_{\mu}}(\boldsymbol{k},\omega) = \kappa_{\mu \nu}(\boldsymbol{k},\omega) E_{\nu}^{\rm ext}(\boldsymbol{k},\omega), \end{equation}

the electrical susceptibility tensor of the system of massless Dirac fermions can be expressed in terms of the Green function as follows:

(5.19)\begin{equation} \kappa_{\mu \nu}(\boldsymbol{k},\omega) = \frac{{\rm i}}{\omega}G_{\mu \nu}(\boldsymbol{k},\omega) , \end{equation}

or, in view of (5.13),

(5.20)\begin{equation} \kappa_{\mu \nu}(\boldsymbol{k}, \omega) = \frac{1}{\hbar \omega V}\int_{0}^{\infty} \exp({-\varepsilon t + {\rm i}\omega t})\left\langle\left[\hat{j}_{\boldsymbol{k}, \mu}(t), \hat{j}_{-\boldsymbol{k}, \nu}(0)\right]\right\rangle\,{\rm d}t \quad (\varepsilon > 0) , \end{equation}

where $\langle \hat {f}(t)\rangle \equiv \textrm {Tr}(\hat {w}\hat {f}(t))$. In equation (5.20) introduced is an infinitesimally small positive quantity $\varepsilon$ for adiabatic switching on of a periodic external electric field. The limit $\varepsilon \downarrow 0$ is taken after the thermodynamic limit $V \rightarrow \infty$ ($V/N = \textrm {const.}$), $N$ is the number of plasma particles.

5.2. Electrical conductivity tensor

Oscillations similar to those of the DHVA effect are observed in kinetic phenomena, for example, in the electrical conductivity. Here we will consider conductivity oscillations of a Dirac plasma embedded in a constant strong magnetic field, i.e. the SdH effect (see e.g. Landau & Lifshitz Reference Landau and Lifshitz1980), using a static conductivity model within the framework of the Kubo linear response theory.

Along with the electrical susceptibility tensor in (5.18), the conductivity tensor $\hat {\sigma }(\boldsymbol {k},\omega )$ can also be defined, as a reaction to the total electromagnetic field in the plasma. In the absence of the spatial dispersion, i.e. at $\boldsymbol {k}\rightarrow \boldsymbol {0}$, for the conductivity tensor $\hat {\sigma }(\omega ) = \lim _{\boldsymbol {k}\rightarrow \boldsymbol {0}}\hat {\sigma }(\boldsymbol {k},\omega )$ the Kubo formula (5.20) is valid (Zubarev Reference Zubarev1974),

(5.21)\begin{equation} \sigma_{\mu \nu}(\omega) = \frac{1}{\hbar \omega V}\lim_{\varepsilon \downarrow 0} \lim_{\boldsymbol{k}\rightarrow \boldsymbol{0}}\int_{0}^{\infty} \exp({-\varepsilon t + {\rm i}\omega t})\left\langle \left[\hat{j}_{\boldsymbol{k},\mu}(t),\hat{j}_{-\boldsymbol{k},\nu}(0)\right]\right\rangle \,{\rm d}t . \end{equation}

Then, the Hermitian part of the electrical conductivity tensor (5.21) of a Dirac plasma in a magnetic field reads

(5.22)\begin{equation} \sigma_{\mu \nu ,{\rm H}}(\omega) = \sigma_{\mu \nu ,{\rm H},+}(\omega) + \sigma_{\mu \nu ,{\rm H},-}(\omega) , \end{equation}

where

(5.23)\begin{equation} \sigma_{\mu \nu ,{\rm H},\pm}(\omega) = \frac{\rm \pi}{\hbar \omega}\sum_{n,m} \left[f_{\rm FD}(\epsilon_{n,\pm}) - f_{\rm FD}(\epsilon_{n,\pm} + \hbar \omega)\right] \left\langle n|j_{\mu}|m\right\rangle_{{\pm}}\left\langle m|j_{\nu}|n \right\rangle_{{\pm}}\delta (\omega - \omega_{m,n,\pm}) \end{equation}

are the conductivity tensors corresponding to right-handed ($+$) and left-handed ($-$) helicities, the subscript $H$ denotes the Hermitian matrix,

(5.24)\begin{equation} f_{\rm FD}(\epsilon_{n,\pm}) = \frac{1}{\exp(\beta \epsilon_{n,\pm} - \eta) + 1} \end{equation}

is the Fermi–Dirac distribution density, which obeys the normalization condition (D2) for the 3-D case and (D12) for the 2-D case, and $\omega _{n,m,\pm } = (\epsilon _{n,\pm } - \epsilon _{m,\pm })/\hbar$. In equation (5.23)

(5.25a,b)\begin{equation} \left\langle n|j_{\mu}|m\right\rangle_{{\pm}} = \int \,{\rm d}\boldsymbol{r} \left\langle n|j_{\mu}(\boldsymbol{r})|m\right\rangle_{{\pm}} , \quad \hat{\boldsymbol{j}}_+{=} -ev\hat{\xi}^+\hat{\boldsymbol{\sigma}}\hat{\xi}, \quad \hat{\boldsymbol{j}}_{-} = ev\hat{\eta}^+\hat{\boldsymbol{\sigma}}\hat{\eta} \end{equation}

are the matrix elements for the right-handed and the left-handed helicities, in the bra-ket (Dirac) notation, of the current-density operator $\boldsymbol {j}(\boldsymbol {r})$.

To account for the scattering of the Dirac fermions in the relaxation time approximation, the $\delta$-function with the energy conservation law in (5.23) should be ‘smeared out’ by replacing the adiabatic parameter $\Delta$ with the collision frequency $\nu _{\pm }$ (Skobov & Kaner Reference Skobov and Kaner1964; Blank & Kaner Reference Blank and Kaner1966). Then

(5.26)\begin{align} \sigma_{\mu \nu, {\rm H},\pm}(\omega) & = \sum_{n,m}\frac{f_{\rm FD}(\epsilon_{n,\pm}) - f_{\rm FD}(\epsilon_{n,\pm} + \hbar \omega)}{\hbar \omega} \nonumber\\ & \quad \times \frac{\nu_{{\pm}}}{(\omega - \omega_{m,n,\pm})^2 + \nu_{{\pm}}^2} \langle n|j_{\mu}|m\rangle_{{\pm}}\langle m|j_{\nu}|n\rangle_{{\pm}} \end{align}

and in the static limit $\sigma _{0,\mu \nu,\pm } = \lim _{\omega \rightarrow 0} \sigma _{\mu \nu,\textrm {H},\pm }(\omega )$, we obtain

(5.27)\begin{equation} \sigma _{0,\mu \nu ,\pm} ={-}\frac{eH}{4{\rm \pi}^2 \hbar c}\int_{-\infty}^{\infty} \sum_{n,m=0}^{\infty}\left. \frac{\partial f_{\rm FD}(\epsilon ,k_{z})}{\partial \epsilon} \right\vert_{\epsilon = \epsilon_{n,\pm}}\frac{\nu_{{\pm}}}{\omega_{n,m,\pm}^{2} + \nu_{{\pm}}^2}\left\langle n|j_{\mu}|m\right\rangle_{{\pm}} \left\langle m|j_{\nu}|n\right\rangle_{{\pm}}\,{\rm d}k_z. \end{equation}

The relaxation time approximation in (5.26) is equivalent to replacing the collision integral in the kinetic equation with the quantity $\sim \delta f /\tau _{\pm }$ (Abrikosov Reference Abrikosov1988), where $\delta f$ is the small correction to the Fermi–Dirac distribution due to collisions, and the relaxation time $\tau _{\pm } = 1/\nu _{\pm }$. The collision frequency $\nu _{\pm }$ is calculated in (5.34).

Using the Dirichlet formula (2.7) in the Supplemental material the expression in (5.27) for the static electrical conductivity tensor can be transformed into

(5.28)\begin{equation} \sigma_{0,\mu \nu ,\pm} ={-}\frac{eH}{4{\rm \pi} ^{2}\hbar c}\int_{-\infty}^{\infty} \sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\left\{\mathcal{F}_{n,n+m,\pm}(k_{z}) + \mathcal{F}_{n+m,n,\pm}(k_{z})\right\}\,{\rm d}k_z, \end{equation}

where $\mathcal {F}_{n,n+m,\pm }(k_{z})$ and $\mathcal {F}_{n+m,n,\pm }(k_{z})$ are given by equation (2.8) in the Supplemental material.

Taking into account of (3.7), after some algebra, the Hermitian part of the static electrical conductivity tensor of a 3-D Dirac plasma in a magnetic field can be represented as

(5.29)\begin{equation} \hat{\sigma}_0 = \hat{\sigma}_{0, +} + \hat{\sigma}_{0, -} , \quad \hat{\sigma}_{0,\pm} = \begin{pmatrix} \sigma _{xx,\pm} & {\rm i}\sigma_{xy,\pm} & 0 \\ -{\rm i}\sigma_{xy,\pm} & \sigma_{yy,\pm} & 0 \\ 0 & 0 & \sigma_{zz,\pm} \end{pmatrix} , \end{equation}

where the Cartesian components $\sigma _{ij,\pm }$ ($i,j = x,y,z$) are given by (C1).

In turn, the static electrical conductivity tensor of a 2-D magnetized massless Dirac plasma takes the form

(5.30)\begin{equation} \hat{\sigma}_0 = \hat{\sigma}_{0,+} + \hat{\sigma}_{0,-} , \quad \hat{\sigma}_{0,\pm} = \begin{pmatrix} \sigma_{xx,\pm} & {\rm i}\sigma_{xy,\pm} \\ -{\rm i}\sigma_{xy,\pm} & \sigma_{yy,\pm} \\ \end{pmatrix} , \end{equation}

where

(5.31)\begin{equation} \left. \begin{aligned} \sigma_{xx, \pm} & ={-}2{\rm \pi} \omega_{\rm H}^2\frac{\hbar^4 v^2}{R_k}\sum_{n = 0}^{\infty} \left[\mathfrak{F}(\epsilon_{n,\pm}) + \mathfrak{F}(\epsilon_{n+1,\pm})\right] \mathcal{J}_{n,n+1,\pm}(x,x,0) , \\ \sigma_{yy, \pm} & ={-}2{\rm \pi} \omega_{\rm H}^2\frac{\hbar^4 v^2}{R_k}\sum_{n = 0}^{\infty} \left[\mathfrak{F}(\epsilon_{n,\pm}) + \mathfrak{F}(\epsilon_{n+1,\pm})\right] \mathcal{J}_{n,n+1,\pm}(y,y,0) , \\ \sigma_{xy, \pm} & ={-}2{\rm \pi} \omega_{\rm H}^2\frac{\hbar^4 v^2}{R_k}\sum_{n = 0}^{\infty} \left[\mathfrak{F}(\epsilon_{n,\pm}) - \mathfrak{F}(\epsilon_{n+1,\pm})\right] \mathcal{J}_{n,n+1,\pm}(x,y,0) \end{aligned} \right\} \end{equation}

and $\epsilon _{n,\pm }$ are given by (3.8a,b). In particular, for a degenerate Dirac plasma ($\beta \epsilon _\textrm {F} \gg 1$), $\mathfrak {F}(\epsilon _{n,\pm }) = - \delta (\epsilon _{n,\pm } - \epsilon _\textrm {F})$ and the 2-D electrical conductivity tensor (5.31) is

(5.32)\begin{equation} \left. \begin{aligned} \sigma_{ii,-} & = 4{\rm \pi} \epsilon_{\rm F} \frac{\hbar^2 v^2}{R_k} \left[\mathcal{J}_{[n_0],[n_0]+1,-}(i,i,0) + \mathcal{J}_{[n_0]-1,[n_0],-}(i,i,0)\right] ,\\ \sigma_{ii,+} & = 4{\rm \pi} \epsilon_{\rm F} \frac{\hbar^2 v^2}{R_k} \left[\mathcal{J}_{[n_0]-1,[n_0],+}(i,i,0) + \mathcal{J}_{[n_0]-2,[n_0]-1,+}(i,i,0)\right] , \quad i = x, y\\ \sigma_{xy,-} & = 4{\rm \pi} \epsilon_{\rm F} \frac{\hbar^2 v^2}{R_k} \left[\mathcal{J}_{[n_0],[n_0]+1,-}(x,y,0) - \mathcal{J}_{[n_0]-1,[n_0],-}(x,y,0)\right] ,\\ \sigma_{xy,+} & = 4{\rm \pi} \epsilon_{\rm F} \frac{\hbar^2 v^2}{R_k} \left[\mathcal{J}_{[n_0]-1,[n_0],+}(x,y,0) - \mathcal{J}_{[n_0]-2,[n_0]-1,+}(x,y,0)\right] , \end{aligned} \right\} \end{equation}

where $[n_0]$ is the integer part of $n_0 = \epsilon _\textrm {F}^2/(\hbar ^2 \omega _\textrm {H}^2)$. For example, for the surface number density of Dirac fermions $n_s = 10^{21}$ cm$^{-2}$ and the magnetic field $H = 30$ MG, $[n_0] = 1$. Then, taking into account that $\omega _\textrm {H} \gg \nu _{\pm }$, we obtain

(5.33a–c)\begin{equation} \sigma_{xx} \simeq \frac{\nu}{\omega_{\rm H}}\frac{59}{R_k} , \quad \sigma_{yy} \simeq \frac{\nu}{\omega_{\rm H}}\frac{61}{R_k} , \quad \sigma_{xy} \simeq \frac{\nu}{\omega_{\rm H}}\frac{57}{R_k} , \end{equation}

where $\nu \simeq \nu _{-} \simeq \nu _+$ and $\sigma _{ij} = \sigma _{ij,-} + \sigma _{ij,+}$ ($i,j = x,y$).

Note, that the conductivity of a ‘plasma consisting of charges with a non-zero mass’ is not necessarily isotropic in the plane perpendicular to the magnetic field. This is a consequence of the fact that the currents $\hat {j}_x$ and $\hat {j}_y$ in (5.25a,b) are expressed in terms of the different Pauli matrices $\hat {\sigma }_x$ and $\hat {\sigma }_y$. However, as follows from (A2), $J_{x,n,-}(k_z) = J_{y,n,-}(k_z)$ and therefore $\sigma _{xx,-} = \sigma _{yy,-}$. In other words, for fermions with negative helicity, the static electrical conductivity is isotropic in the plane perpendicular to the magnetic field. But as it can be seen from (A2), $J_{x,n,+}(k_z)\neq J_{y,n,+}(k_z)$ and consequently for fermions with positive helicity $\sigma _{xx,+} \neq \sigma _{yy,+}$.

To calculate the collision frequency $\nu _{\pm }$ in (C1) and (C2a,b), we shall consider scattering from impurities and for the sake of simplicity we assume it to be isotropic. Within the first Born approximation the collision frequency in the presence of a magnetic field is (Abrikosov Reference Abrikosov1988)

(5.34)\begin{equation} \nu_{{\pm}} = \frac{\omega_{\rm H}^2}{4{\rm \pi}}\frac{n_i|v_i|^2}{\hbar^2 v^2}J_{{\pm}} , \quad J_{{\pm}} ={-}\hbar \sum_n \int \left. \frac{\partial f_{\rm FD}(\epsilon)}{\partial \epsilon} \right|_{\epsilon = \epsilon_{n,\pm}}\,{\rm d}k_z = J_{0,\pm} + J_{{\rm osc},\pm} , \end{equation}

where $n_i$ is the impurity atom number density and $v_i$ is the interaction potential of an electron with an impurity atom. According to Poisson's formula (2.2) in the Supplemental material the sum $J_{0,\pm }$ in (5.34) is given by equation (2.3) in the Supplemental material. In turn, for the oscillatory part of the integral, in the 3-D case, we arrive at the following expression:

(5.35)\begin{equation} J_{{\rm osc},\pm} = \frac{4\beta}{\hbar\omega_{\rm H}^2}{\rm Re}\sum_{n=1}^{\infty}J_{n} , \end{equation}

where

(5.36)\begin{equation} J_n = \int_{-\infty}^{\infty}\int_0^{\infty}\frac{\exp \left(\beta\epsilon - \eta \right)} {\left[\exp \left(\beta \epsilon -\eta \right) + 1\right]^2} \exp \left(\frac{2{\rm \pi} in\epsilon^2}{\hbar^2\omega_{\rm H}^2}\right) \exp \left(-\frac{2{\rm \pi} inv^2 k_z^2}{\omega_{\rm H}^2}\right)\epsilon \,{\rm d}\epsilon \,{\rm d}k_z . \end{equation}

The integration over $k_z$ in (5.36) is carried out using equation (2.4) in the Supplemental material and we obtain

(5.37)\begin{equation} J_{{\rm osc},\pm} = \frac{\beta}{\sqrt{2}\hbar v\omega_{\rm H}}\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}} \int_0^{\infty}\frac{\exp \left(\beta \epsilon -\eta \right)} {\left[\exp \left(\beta \epsilon -\eta \right) + 1\right]^2} \sin \left(\frac{2{\rm \pi} n\epsilon^2}{\hbar^{2}\omega_{\rm H}^2} + \frac{\rm \pi}{4}\right) \epsilon \,{\rm d}\epsilon , \end{equation}

where the dimensionless chemical potential $\eta$ is determined by the condition (D2). If the inequality holds $\eta \gg 1$, we make in this integral the change of variable $y-\eta = \xi$ and using the value of the integral $J_2$ in equation (2.5) in the Supplemental material, from (5.37), we finally have for the oscillatory part

(5.38)\begin{equation} J_{\mathrm{osc},\pm}\simeq 8\sqrt{2}{\rm \pi}^2\dfrac{\eta^2}{\beta^3 \hbar^3\omega_{\rm H}^{3}v} \sum_{n=1}^{\infty }\dfrac{\sqrt{n}\sin \left(\dfrac{2{\rm \pi} \eta^2 n}{\beta^{2}\hbar^{2}\omega_{\rm H}^{2}} +\dfrac{\rm \pi}{4}\right)}{\sinh \left[4{\rm \pi}^2 n\eta /(\beta^{2}\hbar^{2}\omega_{\rm H}^{2})\right]} . \end{equation}

In the case if $\eta \lesssim 1$, the integral in (5.37) can be calculated only numerically.

For 2-D Dirac plasmas, the integral in (5.36) can be written as

(5.39)\begin{equation} J_n = \int_0^{\infty}\dfrac{\exp \left(\beta\epsilon - \eta \right)} {\left[\exp \left(\beta \epsilon -\eta \right) + 1\right]^2} \exp \left(\dfrac{2{\rm \pi} in\epsilon^2}{\hbar^2 \omega_{\rm H}^2}\right)\epsilon \,{\rm d}\epsilon \end{equation}

and $J_{\mathrm {osc},\pm }$ in (5.37) transforms into

(5.40)\begin{equation} J_{\mathrm{osc},\pm} = \dfrac{\beta}{\hbar \omega_{\rm H}^2}\sum_{n=1}^{\infty}\int_0^{\infty} \dfrac{\exp \left(\beta \epsilon -\eta \right)} {\left[\exp \left(\beta \epsilon -\eta \right) + 1\right]^2} \sin \left(\dfrac{2{\rm \pi} n\epsilon^2}{\hbar^2\omega_{\rm H}^2}\right)\epsilon \,{\rm d}\epsilon , \end{equation}

where the dimensionless chemical potential $\eta$ is determined by the condition (D12). If the inequality holds $\eta \gg 1$, we make in this integral the change of variable $y-\eta = \xi$, and in view of the complex integral $J_2$ in equation (2.5) in the Supplemental material, we finally obtain

(5.41)\begin{equation} J_{\mathrm{osc},\pm}\simeq 16{\rm \pi}^2\dfrac{\eta^2}{\beta^3 \hbar^3 \omega_{\rm H}^4} \sum_{n=1}^{\infty}\dfrac{n\sin \left(\dfrac{2{\rm \pi} \eta^2 n}{\beta^2 \hbar^2 \omega_{\rm H}^2}\right)} {\sinh \left[4{\rm \pi}^2 n \eta /(\beta^2 \hbar^2 \omega_{\rm H}^2)\right]} . \end{equation}

The functions in (5.38) and (5.41) oscillate with high frequency, which is a manifestation of the SdH effect in 3-D and 2-D massless Dirac plasmas. Its ‘period’ in the variable $1/H$ is the same as in DHVA effect,

(5.42)\begin{equation} \Delta \left(1/H\right) = 2\mathrm{e}\dfrac{\beta^2 \hbar v^2}{c\eta^2} \end{equation}

and it increases with the Dirac fermion speed as $v^{2}$. The dimensionless chemical potential $\eta$ in (5.42) is determined by the conditions (D2) and (D12). As it can be seen from the above calculations, the SdH effect is a direct consequence of Landau diamagnetism.

Some results of numerical calculations for the functions $\sigma = \sigma _{xx,\pm }, \sigma _{yy,\pm }$, $R_{zz} = (\sigma _{zz,\pm })^{-1}$ for the 3-D massless Dirac plasma, and for $\sigma _\textrm {2D} = \sigma _{xx,\pm }, \sigma _{yy,\pm }$ for the 2-D Dirac plasma as a function of the magnetic field $H$ for various densities and temperatures are shown in figures 3–5. Arbitrary units are used for the conductivity, the magnitude of the magnetic field $H$ is measured in MG, and $n_3$ and $n_2$ are the 3-D and 2-D Dirac fermion density, respectively. The dimensionless chemical potential $\eta$ for these figures is determined by solving (D2) and (D12) numerically. These figures clearly demonstrate the SdH effect in magnetized Dirac plasmas.

Figure 3. The function $\sigma$ versus $H$ for $n_3 = 10^{21}$ cm$^{-3}$, $T = 10$ K and $v \approx c/300$.

Figure 4. The function $R_{zz}$ versus $H$ for $n_3 = 10^{21}$ cm$^{-3}$, $T = 100$ K and $v \approx c/300$.

Figure 5. The function $\sigma _\textrm {2D}$ versus $H$ for $n_2 = 10^{19}$ cm$^{-2}$, $T = 10$ K and $v \approx c/300$.

5.3. Dynamic conductivity tensor. The Drude–Lorentz form

As it is stressed in the Supplemental material (see § 1), in the long-wavelength case both the electrical susceptibility and the internal conductivity are response functions. With $\sigma _\textrm {H}(\omega )$ being the positive-definite Hermitian part (5.23)

(5.43)\begin{equation} \hat{\sigma}_{\rm H}(\omega) = \dfrac{1}{2}\left[\hat{\sigma}^{{\dagger}}(\omega) + \hat{\sigma}(\omega)\right] \end{equation}

of the conductivity tensor $\boldsymbol {\hat {\sigma }}(\omega )$ of a Dirac plasma, we introduce its power frequency moments,

(5.44)\begin{equation} {\hat{{\rm T}}}_{\ell}=\dfrac{1}{\rm \pi}\int_{-\infty}^{\infty}\omega^{\ell}\hat{\sigma}_{\rm H}(\omega) \,{\rm d}\omega , \quad \ell = 0,1 . \end{equation}

The moments (5.44) can be calculated using (5.23), as follows:

(5.45)\begin{equation} \left. \begin{aligned} {\rm T}_{0,\mu \nu} & = \dfrac{1}{\hbar}\sum_{n,m}\dfrac{f_{\rm FD}(\epsilon_{n,\pm}) - f_{\rm FD}(\epsilon_{n,\pm} + \hbar \omega_{m,n,\pm})}{\omega_{m,n,\pm}}\left\langle n| j_{\mu}|m\right\rangle_{{\pm}}\left\langle m|j_{\nu}|n\right\rangle_{{\pm}} ,\\ {\rm T}_{1,\mu \nu} & =\dfrac{1}{\hbar}\sum_{n,m}\left[f_{\rm FD}(\epsilon_{n,\pm}) - f_{\rm FD}(\epsilon_{n,\pm} + \hbar \omega_{m,n,\pm})\right] \left\langle n|j_{\mu}|m\right\rangle_{{\pm}}\left\langle m|j_{\nu}|n\right\rangle_{{\pm}} . \end{aligned} \right\} \end{equation}

We wish to reconstruct the tensor $\hat {\sigma }(z)$ for $\textrm {Im}\ z \geq 0$ as a non-canonical solution of the truncated matrix Hamburger moment problem involving only two moments (5.45). Using the matrix version of the Nevanlinna theorem (Kovalishina (Reference Kovalishina1983), Adamyan & Tkachenko (Reference Adamyan, Tkachenko and Verlag2000) and also see appendix B), we arrive at

(5.46)\begin{equation} \hat{\sigma}(z) = \hat{\sigma}_0\left({\hat{{\rm I}}} + {\rm i}{\hat{\varOmega}}_1\hat{\tau} \right)\left[{\hat{{\rm I}}}-{\rm i}\left(z{\hat{{\rm I}}}-{\hat{\varOmega}}_1\right)\hat{\tau}\right]^{{-}1} , \end{equation}

where

(5.47)\begin{equation} {\hat{\varOmega}}_1 = {\hat{{\rm T}}}_0^{{-}1}{\hat{{\rm T}}}_1 . \end{equation}

The dynamic conductivity tensor (5.46) is expressed in terms of the static conductivity tensor $\hat {\sigma }_0$ in (5.29) or (5.30) and the relaxation time tensor $\hat {\tau } = \hat {\nu }^{-1}$ ($\hat {\nu }$ is the collision frequency tensor). In the simplest approximation we can put $\hat {\nu } = {\hat {\textrm {I}}} \nu _{\pm }$, where $\nu _{\pm }$ is given by (5.34). Thus, the dynamic conductivity tensor will also oscillate with high frequency, which is a manifestation of the SdH effect in magnetized massless Dirac plasmas.

The dielectric tensor can be constructed and the spectrum of collective excitations in the system under scrutiny can be studied, but this problem is beyond the scope of the present paper.